Question

if Z is a complex number such that,|z-1|=1 then arg(1/z-1/2)may be




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Answer 1

Given:

|Z-1|=1

To Find:

$arg(\frac{1}{Z} -\frac{1}{2} )$

Solution:

Let Z is a complex number

Z= x +iy

Modulus of complex number:

$|Z|= \sqrt{x^{2} + y^{2} }$

Argument of a complex number:

$arg(z) = Tan^{-1}\frac{y}{x}$

given |z-1|=1

z= x + iy

$|(x-1) + y| = \sqrt{(x-1)^{2} +y^{2} } \\\\x^{2} + y^{2} -2x +1 =1\\ x^{2} + y^{2} = 2x$

$arg(\frac{1}{Z} -\frac{1}{2} )$

$\frac{1}{z}- \frac{1}{2} \\\frac{1}{x+iy}- \frac{1}{2} \\\\\frac{x-iy}{x^{2}+ y^{2} }- \frac{1}{2} \\\\x^{2} + y^{2} =2x \\\frac{x-iy}{2x }- \frac{1}{2} \\\\\\\\\\frac{x}{2x} - \frac{y}{2x} -\frac{1}{2}\\ -\frac{iy}{2x}$

since the function do not contain any real part . hence real part = 0

argument = $Tan^{-1} \frac{\frac{y}{2x} }{0}$

Argument = $\frac{\pi }{2}$